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SageMath
E = EllipticCurve("bh1")
E.isogeny_class()
Elliptic curves in class 12870.bh
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12870.bh1 | 12870bg1 | \([1, -1, 1, -3805973, 2953734581]\) | \(-225817164626811885218547/8821617915914375000\) | \(-238183683729688125000\) | \([3]\) | \(544320\) | \(2.6790\) | \(\Gamma_0(N)\)-optimal |
12870.bh2 | 12870bg2 | \([1, -1, 1, 18669337, 9273598417]\) | \(36561089342650869429237/23047447204589843750\) | \(-453642903327941894531250\) | \([]\) | \(1632960\) | \(3.2283\) |
Rank
sage: E.rank()
The elliptic curves in class 12870.bh have rank \(0\).
Complex multiplication
The elliptic curves in class 12870.bh do not have complex multiplication.Modular form 12870.2.a.bh
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.